The de Rham-Fargues-Fontaine cohomology

Abstract

We show how to attach to any rigid analytic variety $V$ over a perfectoid space $P$ a rigid analytic motive over the Fargues-Fontaine curve $\mathcal{X}(P)$ functorially in $V$ and $P$. We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasi-coherent sheaves over $\mathcal{X}(P)$, and we show that its cohomology groups are vector bundles if $V$ is smooth and proper over $P$ or if $V$ is quasi-compact and $P$ is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit ${\mathbb{B}}^1$-homotopies, the motivic proper base change and the formalism of solid quasi-coherent sheaves.

arXiv commentMinor edits. Published version

References in corpus (7)

• Relative p-adic Hodge theory: Foundations
• Geometrization of the local Langlands correspondence
• A motivic version of the theorem of Fontaine and Wintenberger
• The six-functor formalism for rigid analytic motives
• Crystalline cohomology and de Rham cohomology
• Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve
• Pseudocoherent and Perfect Complexes and Vector Bundles on Analytic Adic Spaces

Cited by in corpus (1)

• The six-functor formalism for rigid analytic motives